Rota-Baxter Operators Of Weight One And Phase Spaces Associated To Left-Symmetric Algebras And Hom-Anti-Flexible Algebras

In this thesis,we mainly give a study on the three topics.Firstly,we compute the Rota-Baxter operators of weight one on some four-dimensional pseudo-Riemannian left-symmetric algebras.And we construct some new left-symmetric algebras by the Rota-Baxter operators under the associative condition.Secondly,we construct some non-abelian eight-dimensional symplectic Lie algebras which called phase spaces by the symmetric solutions of S-equation in a four-dimensional left-symmetric algebra.Finally,we introduce two new kinds of Hom-algebra structure:Hom-anti-flexible algebras and Hom-pre-anti-flexible algebras.Moreover,we study the bimodule of Hom-anti-flexible algebras and give the relationship between such two Hom-algebras by the Hom-O-operator of Hom-anti-flexible algebras.